An equivalent set is a mathematical concept referring to two sets that have the same cardinality, which means they contain the same number of elements. The elements themselves may differ, but the sets are considered equivalent as long as their sizes are the same.
Example 1:
- Set A = {1, 2, 3}
- Set B = {a, b, c}
Since both A and B have 3 elements, they are equivalent.
Example 2:
- Set C = {x, y}
- Set D = {apple, banana}
Since both C and D have 2 elements, they are equivalent sets.
Properties of Equivalent Sets
Important properties of equivalent sets are
- Cardinality: If two sets have the same number of elements, they are equivalent.
- No Need for Identical Elements: The specific elements of the sets do not matter, only the count.
- One-to-One Correspondence: If you can establish a one-to-one correspondence (bijective mapping) between the elements of two sets, the sets are equivalent.
Equal vs Equivalent Set
Key differences between equal and equivalent sets are:
| Aspect | Equal Set | Equivalent Set |
|---|---|---|
| Definition | Two sets are equal if they have the exact same elements. | Two sets are equivalent if they have the same number of elements. |
| Element Comparison | Elements in both sets must be identical, regardless of order. | Elements are not compared, only the count matters. |
| Symbol | Denoted as A = B. | Denoted as A ∼ B. |
| Example of Sets | A = {1, 2, 3}, B = {1, 2, 3} → A = B | A = {1, 2, 3}, B = {a, b, c} → A∼B |
| Focus | Focuses on the elements themselves. | Focuses on the cardinality (number of elements). |
| Relationship | All equal sets are equivalent sets. | Not all equivalent sets are equal sets. |
How Do You Check if Two Sets Are Equivalent
To check if two sets are equivalent:
- Count the number of elements in both sets.
- If the counts are equal, the sets are equivalent.
Practice Questions
Q1. Let A = {x: x is a letter in the word "ALGEBRA"} and B = {1,2,3,4,5,6}.
Q2. For what value of n are the following two sets equivalent?
A = {x: x ∈ N , x ≤ n} and B = {2,4,6,…,20}.
Q3. Set A has 12 elements and set B has 12 elements. Set C has 11 elements. Which pairs among A,B,C are guaranteed to be equivalent? Explain why.
Q4. Can two sets be equivalent even if they have no common elements? Give an example and explain your answer.
Q5. A set A contains all the factors of 120, while set B contains all the factors of 180. Are A and B equivalent.